572
10 Quantum Statistics
We have seen that, quite generally, the internal energy U can be obtained from
the grand partition function as
U = −
∂
∂β
ln
λ,V
= k B T
2
∂
∂T
P V
k B T
λ,V
and, as
P V
k B T
BE
=
1
3 (T )
g
−
5
2
(λ)
for T > T c , we obtain U as
U = k B T
2 g
−
5
2
(λ)
d
dT
1
3
=
3
2 k B T
V
3 g
−
5
2
(λ)
for T > T c . To obtain U for T < T c , we need only replace λ by 1, so that the two
expressions for the internal energy can be merged to give
U =
⎧
⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎩
3k B T V
2 3 g
−
5
2
(λ) , T > T c
k B T
3 g
−
5
2
(1) , T < T c
.
(10.6.29)
The entropy for a BE gas can readily be obtained from the thermodynamic
relation Nμ = U − T S + P V as
S
Nk B
=
U + P V
Nk B T
−
μ
k B T
.
We thus find that the entropy is given as
S
Nk B
=
⎧
⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎩
5V
2NN 3 g
−
5
2
(λ) − ln λ , for T > T c
5V
2NN 3 g
−
5
2
(1) ,
for T < T c
.
(10.6.30)
Finally, the heat capacity at constant volume of a Bose–Einstein gas can be
obtained as